Search Results (5)

This study is devoted to deriving precise pointwise estimates for the linear Cauchy problem of the wave equation with scale-invariant damping. The proof relies on representing the solution via Fourier integral operators and employing techniques from microlocal analysis. As an application, we establish weighted Strichartz estimates for the linear system. Full article

Time-of-flight magnetic resonance angiography (TOF-MRA) is widely used for noninvasive visualization of arterial anatomy, but extracting hemodynamics like blood velocity typically requires supplementary phase-contrast scans, tagging or multi-TE images. This study proposes a novel, physics-informed computational framework to extract variable fluid velocity directly from standard TOF-MRA signal profiles. We analytically expand the approach-to-steady-state Bloch equations to include convective flow, establishing a mathematical relationship between the spatial decay of longitudinal magnetization and fluid velocity. The velocity derivation was further extended to pointwise estimation over a 1-D centerline, overcoming the limitations of constant-velocity assumptions. To validate and solve this problem, a MATLAB (R2025b) simulation framework was developed to model fluid flow in two variable-geometry flowing tube cases, i.e., continuous narrowing and focal stenosis, under synthetic scanner noise. A global inverse optimization approach utilizing Dual-Tikhonov regularization was applied to stably invert the ill-posed transit time integral, actively penalizing high-frequency numerical ringing while preserving structural curves. The computational simulations successfully recovered ground-truth point-wise velocities, tracking gradual hemodynamic accelerations and sharp stenotic jets. This theoretical framework and the example centerline TOF-MRA signal intensity provide a robust mathematical proof-of-concept that quantitative, localized functional hemodynamic metrics can be extracted from standard structural MRA imaging, establishing a foundation for advanced flow quantification without requiring additional scan time. Full article

In this paper, we derive the global dynamic properties of the Dirac–Klein–Gordon system in ${\mathbb{R}}^{1+2}$ with a class of large initial data. We consider the case of a massless Dirac field coupled with a massive Klein–Gordon field. The initial data are bounded in certain weighted Sobolev spaces, where the Dirac field is small while the Klein–Gordon field can be arbitrarily large. We establish global existence and characterize the asymptotic behavior of the solutions, including sharp time decay and linear scattering. Full article

We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functions f on the unit cube $I^2$ in ${\mathbb{R}}^2$ . Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) of the expansion of f in the rectangular Schauder system, near the point considered. We deduce that pointwise rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities. As an application, we refine earlier results in Ayache et al. (Drap brownien fractionnaire. Potential Anal. 2002, 17, 31–43) and Kamont (On the fractional anisotropic Wiener field. Probab. Math. Statist. 1996, 16, 85–98), where uniform rectangular Lipschitz regularity of the trajectories of the fractional Brownian sheet over the total $I^2$ (or any cube) was considered. Actually, we prove that fractional Brownian sheets are pointwise rectangular and level coordinate axes monofractal. On the opposite, we construct a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal. Full article